Partial Fraction Decompositions

If we factor out the denominator, we get

$x^{2} - 8x - 7$ = $(x - 1) (x - 7)$

So, our denominator of the expanded fractions must be $x - 1$ and $x - 7$ .

Let out numerators be $A$ and $B$ .

We get

$17x - 71 = A(x - 7) + B(x - 1)$

We set $x$ at different values where we can find values of $A$ and $B$ . We eliminate the other unknown by multiplying it with a value of $0$ .

When $x = 1$

$17(1) - 71 = A(1 - 7) + B(1 - 1)$

$A = 9$

When $x = 7$

$17(1) - 71 = A(7 - 7) + B(7 - 1)$

$B = 8$

In conclusion,